No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

Kinetic stability theorem for relativistic non‐neutral electron flow in a planar diode with applied magnetic field

### Abstract

A kinetic stability theorem is developed for relativistic non‐neutral electron flow in a planar high‐voltage diode with applied magnetic field. The effects of strong inhomogeneities and intense self‐electric and self‐magnetic fields are retained in the analysis in a fully self‐consistent manner. Use is made of global (spatially averaged) conservation constraints satisfied by the fully nonlinear Vlasov–Maxwell equations, assuming electromagnetic perturbations with extraordinary‐mode polarization, and space‐charge‐limited flow with *E* ^{0} _{ x }(*x*=0)=0 at the cathode. It is also assumed that the *y*‐averaged, *x*‐directed net flux of particles, *y* momentum, and energy, vanish identically at the cathode (*x*=0) and at the anode (*x*=*d*). It is shown that the class of self‐consistent Vlasov equilibria *f* ^{0} _{ b }(*H*,*P* _{ y }) is stable for small‐amplitude perturbations, provided *f* ^{0} _{ b } is a monotonic decreasing function of *H*−*V* _{ b } *P* _{ y }, i.e., provided ∂*f* ^{0} _{ b }/∂(*H*−*V* _{ b } *P* _{ y })≤0. Here, *H* is the energy and *P* _{ y } is the canonical *y* momentum. The generality of this *s* *u* *f* *f* *i* *c* *i* *e* *n* *t* *c* *o* *n* *d* *i* *t* *i* *o* *n* *f* *o* *r* *s* *t* *a* *b* *i* *l* *i* *t* *y* should be emphasized. First, the derivation of the stability theorem has not been restricted to a specific choice of *f* ^{0} _{ b }(*H*−*V* _{ b } *P* _{ y }). Moreover, the fully non‐neutral electron equilibria are generally characterized by strong spatial inhomogeneities and intense self‐electric and self‐magnetic fields. For the class of equilibria with ∂*f* ^{0} _{ b }/∂(*H*−*V* _{ b } *P* _{ y })≤0, it is also shown that the density profile *n* ^{0} _{ b }(*x*)=∫ *d* ^{3} *p* *f* ^{0} _{ b } and *x*–*x* pressure profile *P* ^{0} _{ b }(*x*) =∫ *d* ^{3} *p* *v* _{ x } *p* _{ x } *f* ^{0} _{ b } decrease monotonically from the cathode (*x*=0) to the anode (*x*=*d*) provided the applied magnetic field at the anode (*B* _{ a }) is sufficiently strong that (*V* _{ b }/*c*)*B* _{ a } ≥4π*e* ∫^{ d } _{0} *d* *x* ’ *n* ^{0} _{ b }(*x* ’).

© 1985 American Institute of Physics

Received 01 August 1983
Accepted 30 August 1984

/content/aip/journal/pof1/28/1/10.1063/1.865159

http://aip.metastore.ingenta.com/content/aip/journal/pof1/28/1/10.1063/1.865159

Article metrics loading...

/content/aip/journal/pof1/28/1/10.1063/1.865159

1985-01-01

2016-09-26

Full text loading...

Commenting has been disabled for this content